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Questions tagged [groupoids]

A groupoid is a small category in which every morphism is an isomorphism. They arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag. Also, use other tags such as monoid or category-theory, if needed.

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What is the canonical action of a groupoid on its unit space?

Supposedly, a groupoid $G$ acts canonically on its unit space $G^{(0)}$. What is this action explicitly? I think this is the action where an arrow takes its source unit element to the target/range of ...
Panini's user avatar
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Bijection involving the fundamental groupoid of a manifold

Let $M$ be a smooth manifold. I read in this post, that there is a bijection between the fundamental groupoid $\Pi(M)$ and $(\tilde{M}\times \tilde{M})/\pi_1(M)$, where $\tilde{M}$ is the universal ...
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Lattices of Lie Groupoids

There exists an important concept in Lie group theory being that of lattice. Let $G$ be a Lie group, a lattice $\Gamma$ is a discrete subgroup $\Gamma \subseteq G$ such that the quotient $G/\Gamma$ ...
Tomás Pacheco's user avatar
2 votes
1 answer
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Why is this localization thin?

I am studying this paper by Malkiewich and Ponto. I am unsure about one claim. Let $\Delta$ be the augmented simplex category. Denote by $\mathfrak{J}$ the wide subcategory of $\Delta$ consisting of ...
Learner's user avatar
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Characterization of the Predual of a von Neumann Algebra

I was reading J. Renault's paper "The Fourier Algebra of a Measured Groupoid" and I am confused about his approach to the predual of a von Neumann Algebra. Let $M:= VN(\mathcal{G})$ be the ...
Tomás Pacheco's user avatar
1 vote
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33 views

Definition of equivalence of topological groupoids and cohomology of groupoids

When we have a discrete groupoid, we have the concept of equivalence of groupoids as categories. Given two equivalent discrete groupoids, we obtain a simplicial homotopy between their respective ...
Emmanuel Jerez's user avatar
3 votes
1 answer
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Equality of norms Groupoid $C^*$-algebra

In the post " Reduced groupoid $C^*$-algebra " I discuss two possible ways to construct the reduced groupoid $C^* $-algebra $C_\lambda^* (G)$ of an étale locally compact Hausdorff groupoid $...
Tomás Pacheco's user avatar
6 votes
1 answer
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Reduced groupoid $C^*$-algebra

I was reading Ozawa's book and in Chapter 5 they discuss $C^* $-algebras of locally compact Hausforff étale groupoids. I have a question regarding the reduced $C^* $-algebra construction. It is first ...
Tomás Pacheco's user avatar
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Showing fibers of an r-discrete groupoid are discrete

I'm working off of the definition that $\mathcal{G}$ is a groupoid if it has a partial product which is associative (when applicable) and an involutive inverse map so that $xx^{-1}$ is always defined ...
Joseph DeGaetani's user avatar
3 votes
1 answer
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Left adjoint to forgetful functor from groups to groupoids, generalizing injective inclusions to free product of groups

Is there a left adjoint $F$ to the "forgetful" inclusion functor $U$ from the category of groups (interpreted as groupoids with one object $*$) to the category of groupoids? If so, then ...
I Eat Groups's user avatar
4 votes
2 answers
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Recovering a topological space from its fundamental groupoid [closed]

Given only the fundamental groupoid of a topological space X, we can recover the underlying set of X since objects of the groupoid (as a category) are precisely the elements of X. I would like to know ...
toby flenderson's user avatar
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Is the arrow space of a fundamental groupoid of a Hausdorff and second countable manifold always Hausdorff and second countable?

Usually in the definition of a Lie groupoid, we do not assume the arrow space to be second countable and Hausdorff. Now, in particular for a smooth manifold $M$ with the assumption that it is second ...
mathematics student's user avatar
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Crossed module correspondence to double groupoid

Reading Brown & Spencers papper on double groupoids and crossed modules I don't quite understand the constructed double groupoid(see http://www.numdam.org/item/CTGDC_1976__17_4_343_0.pdf page 11) ...
Mrsalladhead 's user avatar
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Confusion in the definition of Algebraic structure, system, operation, magma

During studying the text book of abstract algebra by john Farleigh, I encounter with tye definition of binary Algebraic structure. Then I tried to find the difference between binary Algebraic ...
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Reference about connected groupoids and action groupoids

I need a reference for the assertion that: "any connected groupoid is isomorphic to some action groupoid (of a group action on the set of objects of the groupoid)", as discussed in this ...
Emmanuel Jerez's user avatar
9 votes
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Fundamental groupoid of a filtered union

Let $X$ be a topological space and let $(X_i)_{i\in I}$ be a filtered family of subspaces. Let $X =\bigcup_{i \in I} X^°_i$ be the union of the interiors of the $X_i$. I want to prove the following ...
Alice in Wonderland's user avatar
5 votes
1 answer
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Functor $B: \mathsf{Group} \to \mathsf{Groupoid}$ preserves pushout

I came across a lemma that says that the functor $B: \mathsf{Group} \to \mathsf{Groupoid}$ preserves pushout, where, for a group $G$, $BG$ is just the groupoid with one object $\{ \star\}$ and ...
Alice in Wonderland's user avatar
3 votes
1 answer
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Is every groupoid $\mathcal{G}$ uniquely represented by a group and cardinality of $\mathcal{G}_{Obj}$?

Edit: as user8268 noticed, what follows refers to connected groupoids only. I have intuition, that while groupoids in general might have rich internal structure, all finite groupoids are "trivial&...
Przemek's user avatar
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Existence of the category of groupoids

I am aware of groupoid categories. Recently, I came across the category of all the groupoids in JP May's A Concise Course in Algebraic Topology, wherein he describes: Taking morphisms to be functors, ...
Atom's user avatar
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Definition of $G$-set in the context of groupoids, Renault

In Renault's A Groupoid Approach to $C^*$-Algebras, the following definition is given: Let $G$ be a groupoid. A subset $S$ of $G$ will be called a $G$-set if the restrictions of $r$ [the range map] ...
MakeOperatorAlgebrasGreatAgain's user avatar
2 votes
1 answer
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Morphisms from simplicial sets into groupoids are determined by what?

For this question, one can assume that the simplicial sets come from oriented simplicial complexes (you can assume the orientation induces an orientation on the geometric realization). A morphism from ...
JLA's user avatar
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1 vote
2 answers
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A name for morphisms with the same source object

Let $G$ be a groupoid (or, more generally, a category). Given an object $o \in \mathrm{Obj}(G)$, is there a name for the set of morphisms having $o$ as their source object? The motivation is the ...
Seirios's user avatar
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Name of a groupoid

Let $S$ be an arbitrary set. Consider the category whose objects are the points of $S$; whose morphisms are the pairs $(a,b) \in S^2$, with starting and ending objects respectively $a$ and $b$; such ...
Seirios's user avatar
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Why is it worth studying groupoids if they are so similar to groups?

I know that in the pure algebraic context, connected groupoids are equivalent to groups. Moreover, connected groupoids can be seen as action groupoids. Consequently, any abstract groupoid is ...
Emmanuel Jerez's user avatar
2 votes
1 answer
96 views

is the orbit space of a proper (but non-open) topological groupoid always Hausdorff?

I feel this ought to be a very simple question, but I seem to be asking it at a greater level of generality than people usually do: Is the orbit space $|X|$ of every proper topological groupoid $G \...
Chris Wendl's user avatar
11 votes
1 answer
263 views

Conjugation Functor from a Groupoid to $\mathbf{Grp}$

Take a groupoid $\mathcal{C} \in \mathbf{Grpd}$. It's possible to construct a conjugation functor $F_{\mathcal{C} } : \mathcal{C} \to \mathbf{Grp}$ as follows: For every object $x \in \text{ob}(\...
Rubaiyat Khondaker's user avatar
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When is projection from a group-scheme-action étale?

I am relatively inexperienced with schemes and I am having trouble finding references to results specifying when the projection morphism from a fiber product with a group scheme is étale. More ...
Absent mind's user avatar
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Proof of Square Zero Property for the Total Differential in the de Rham Double Complex of a Lie Groupoid

I am trying to understand the de Rham double complex associated to a Lie groupoid, and I am having trouble proving a fundamental property of the total differential, i.e., that it squares to zero. ...
Nash's user avatar
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Proving isotropy of a groupoid multiplication graph in a quasi-presymplectic groupoid

I'm studying quasi-presymplectic groupoids and I've come across the following proposition which I'm finding challenging to prove. Any help or guidance would be greatly appreciated. Given a Lie ...
Carl's user avatar
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1 answer
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Categories fibered in groupoids and the slice category

I read that one example of categories fibered in groupoids is the slice category $\mathcal{C}_{/x}\to \mathcal{C}$ where $x\in \mathcal{C}$ an object. But as the usual definition of slice category ...
Chanel Rose's user avatar
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geometric meaning of viewing $G$-cover of $X$ as a functor $\Pi_1(X) \to BG$

Let $G$ be a finite discrete group. A $G$-cover on $X$ is a covering space $p: E \to X$ with group of deck transformations $G$ acting transitively on each fiber (so a principal $G$-bundle). In Qiaochu ...
Tanny Sieben's user avatar
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Every Category with a Zero Object contains a Connected Groupoid.

Let $C$ be a Small Category containing a Zero Object $Z$. $(\forall{X,Y}\in(Obj(C)))$ define the Zero Morphism $\psi_{X,Y}:X\rightarrow{Y}$ as the composition $\psi_{X,Y}=I_{Y}\circ{T_{X}}$ where $T_{...
user640930's user avatar
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Why do we have $s.\chi_x=\chi_x$ implies that $s(x)=x u$ for some $u \in \mathfrak{C}^{*, 0}$ in an inverse semigroup?

This is a detail in Theorem 4.4 on Page 9 of Xin Li's paper Left Regular Representations of Garside Categories I. C-star-Algebras and Groupoids. In the proof he said that: $$s.\chi_x=\chi_x \text{ ...
ScienceAge's user avatar
3 votes
1 answer
90 views

What categorical notion of "equivalence" does $B:\text{Grp}\to \text{ConnGrpd}$ give?

I've been taking a course on Category Theory recently and really enjoying it. The lecturer today was discussing the notion of equivalence in more detail and showed that every connected groupoid $\...
Isky Mathews's user avatar
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4 votes
1 answer
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$(a^{-1})^{-1} = a$ in a groupoid: proof review

Exercise on groupoids article: Show that in a groupoid, $$ (a^{-1})^{-1} = a \,. $$ Proof: Every element must have an inverse, and likewise each element’s inverse must itself have an inverse (e.g., $...
Hank Igoe's user avatar
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1 vote
1 answer
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Functors between groupoids

I want to prove that given two groupoids $G$ and $H$, a "functor candidate" $F:G \to H$ between them needs only to preserve compositions, that is, in the case of groupoids, to be a functor ...
André's user avatar
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Understanding the "fundamental theorem of covering spaces"

I ask this question because I'm quite confused about the fundamental theorem of covering spaces ; This theorem say that under suitable hypothesis for a topological space $X$ (semi locally simply ...
Emile oleon's user avatar
1 vote
0 answers
71 views

Why is groupoids the appropriate tool for studying quasicrystals?

According to Wikipedia, groupoids is the appropriate tool for studying quasicrystals. Classical theory of crystals reduces crystals to point lattices where each point is the center of mass of one of ...
Changsu Wang's user avatar
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1 answer
75 views

What is a pointed connected groupoid?

Here it says a pointed connected groupoid is a connected groupoid that is pointed. This seems nonsensical: connected groupoids don't have a terminal object, except for the trivial case. The link in ...
Carla only proves trivial prop's user avatar
4 votes
1 answer
232 views

Why, conceptually, isn't the 2-category of connected groupoids equivalent to the 2-category of groups?

It is well-known that every connected groupoid is equivalent to a group. However the 2-category of connected groupoids is not equivalent to the 2-category obtained trivially from the 1-category of ...
Carla only proves trivial prop's user avatar
3 votes
1 answer
102 views

Groupoid embedding into a group [closed]

A necessary condition for a groupoid $(G,∗)$ to embed into a group is that for all $a,b\in G$ then $a ∗ a^{-1}=b∗b^{−1}$. Question: Is this necessary condition also sufficient? This question came to ...
Sebastien Palcoux's user avatar
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1 answer
112 views

How to prove that a unique inverse exists for every element of this group like structure

Given a quartet $(s,p,m, n)$ and a law $$(s,p,m,n).(q,r,t,u)=\left(\frac{2}{3}sq, pr,m+(1+t), nu\right)$$ where $s,p,m,n,q,r,t$ and $u$ are in $\mathbb{R} \setminus \{0\}$. The right and left ...
cows's user avatar
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1 vote
1 answer
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Proving a group like object always has a unique right inverse, and a unique left inverse

Given a quartet $(s,p,m, n)$ and a law $$(s,p,m,n).(q,r,t,u)=(\frac{2}{3}sq, pr,m+(1-t), nu)$$ where $s,p,m,n,q,r,t$ and $u$ are in $\mathbb{R} \setminus \{0\}$. The left and right identities have ...
cows's user avatar
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3 votes
1 answer
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Name for the group generated by a groupoid

Is there a name for a group generated by a groupoid? I'm defining a groupoid as a category $\mathcal{C}$ where every arrow has a unique inverse that is both a left inverse and a right inverse. I'm ...
Greg Nisbet's user avatar
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Obtaining $\mathscr{B}G$ from the topological groupoid $BG$; which notion of "nerve" of a topological groupoid/category should be used?

For $G$ a group without topology (or a discrete topological group), let $BG$ denote the groupoid with one object and morphisms given by $G$. Then, as described at this nLab page, the geometric ...
I.A.S. Tambe's user avatar
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2 votes
2 answers
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The monoidal structure on the fundamental groupoids of spectrum

I am trying to understand Anderson duality and Picard categories from appendix B of Hopkins and Singer's paper, and I somehow get stuck on Example B.7 (Page 87). For a spectrum $E$, they consider each ...
timaeus's user avatar
  • 321
3 votes
3 answers
191 views

Categorical Intuition of Path Induction

I am trying to understand path induction from the trinitarian point of view. So far I understand the informal intuition of path induction from a homotopical and computational point of view. But I am ...
IsAdisplayName's user avatar
2 votes
0 answers
36 views

Continuity of Norm of Regular Representations on Étale Groupoids

Let $\mathcal G$ be a locally compact, Hausdorff, étale groupoid. Following the notation found in Sims' notes (see here), for $x\in\mathcal G^{(0)}$, let $\lambda_x:C_c(\mathcal G)\to\mathbb B(\ell^2(...
Aweygan's user avatar
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6 votes
0 answers
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Is the Mathieu Groupoid $M_{13}$ even special, besides its construction?

The Mathieu groupoid $M_{13}$ has a beautiful construction from a "sliding block puzzle" produced from the projective plane of order $3$; put $12$ labelled counters on $12$ of the $13$ ...
Beren Gunsolus's user avatar
4 votes
1 answer
183 views

A groupoid that is not a fundamental groupoid

There is a nice result that says for any group $G$, there exists a topological space $X$ such that the fundamental group $\pi_{1}(X)$ of $X$ is isomorphic to $G$. The proof is not terribly complex: we ...
ckefa's user avatar
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